Assessment of Turbulence Modelling for Cfd

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8/6/2019 Assessment of Turbulence Modelling for Cfd
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ASSESSMENT OF TURBULENCE MODELLING FOR CFDSIMULATIONS INTO HYDROTURBINES: SPIRAL CASINGS
Luís C. Eduardo Oliveira de Souza, Marcelo Dias de Moura, Antonio C. P. Brasil JuniorUniversidade de Brasília  Departamento de Engenharia Mecânica  PósGraduaçãoLaboratório de Energia e Ambiente  LEACEP 70.910900  Brasília  DF  Brasil+55 (61) 3072314 R236/[email protected]
Håkan NilssonDepartment of Thermo and Fluid DynamicsChalmers University of TechnologyGöteborg, [email protected]
Abstract. Spiral casings are applied to distribute the water, as evenly as possible, to the stay vanes and wicket gates and then to theturbine runner. In a welldesigned spiral casing, the pressure head of the fluid should be made available to the runner withminimum loss, hence the analysis of the flow through a spiral casing is important for the design of efficient hydraulic turbines. The
goal of this work is based on turbo machines spiral casings threedimensional flow modeling, simulation and characterization.Governing equations related to their study are used in the mathematical modeling part. Therefore, several turbulence modelsavailable will be tested with a finite volume method based commercial software – CFX 5.5.1 which generates default tetrahedral meshing. Qualitative and quantitative results validation are proved by workshop benchmark experiments already done, in order these results can obtain a turbulence model that represents better the flow complexity inside spiral casings.
Keywords: Numerical simulation, spiral casings, CFD, Turbulence Models.
1. Introduction
Until a few years ago, hydromachines were built only using design methodologies based in empirical relationsaccumulated on two centuries of turbomachinery project experience. The enhancement of the hydroturbine efficiencyon the last decade could only be reached by the use of modern design techniques based in Computational FluidDynamics (CFD). The progress of internal hidrodynamic design, using CFD, is a reality in modern hydroturbines
projects, decreasing the time and costs for the project development. The use of CFD tools is an efficient andinexpensive way to make internal flow predictions accurately, and any sort of flow problems can be detected and further improvements can be made on the geometry component. Some articles have been published in the literature concerningthe use of CFD as a tool of hydroturbine design (Maji and Biswas, 1998 and Maji and Biswas, 1999).
The commercial codes use a lot of alternatives for the modeling and simulation of internal flow problems. Manyturbulence models could be used and different options of boundary conditions (wall laws, outflow, etc.) can be set. Theassessment of turbulence modeling is a great contribution in order to verify the good alternatives of turbulence closuremodels, which reproduces a good agreement to the reality of the fluid flow. The main objective is to reproducerealistically the main structures of the threedimensional fluid flow with a good estimative of pressure drop insideturbine components and torque on the runner surface. What kind of turbulence model works well to do this? Of course it
is a difficulty problem because at each turbine component has specifically flow conditions which are much adaptable toa different alternative of turbulence model. The characteristic of the rotating flow through the runner and the flow insidethe draft tube have important differences which are much treatable to some turbulence modeling strategies.
The aim of this paper is to evaluate three different models of turbulence (k ε, k ω and one equation model KE1E)for the use in the simulations of the spiral casing flows.
Usually in Kaplan or Francis turbines, spiral casings are employed to conduct water from the turbine entrance to thedistributor. Built with complex geometry forms, a good spiral casing is projected to make sure that partial discharges
pass through the guide vane inlet equally. To do this, its sections must gradually decrease towards the flow direction.Therefore, the main difficulties in a spiral casing project are to design such component to envoy fluid to the runner
blades with minimum pressure head loss. Also, suggestions take as the generation of secondary flow what causes loss,Kurokawa and Nagahara (1986).
Flow analysis available in the open literature manipulates different methodologies for numerical simulationdevelopment. Maji and Biswas (1998) combined Petrov Galerkin finite element method to describe the flow structure in
the spiral casing of a reaction turbine. L. Gros et all (1998), used the finite volume based commercial CFD codeTASCflow and Nilsson and Davidson (2001), presented computational results with the finite volume CALCPMB CFDcode.
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The simulation developed in the present paper is performed using the AEA Technology – CFX 5.5.1 commercial program. A spiral casing of Kaplan turbine model was used for present flow analysis. Three turbulence models – included on the code – have been tested to be compared with investigations already observed. Experimental results were
performed by H. Nilsson et all (2001), on a project with the Swedish National Energy Administration and GE Energy(Sweden) AB, which supplied the geometry drawing and the measurements for comparison and code validation.
The hydraulic component was a Kaplan turbine semispiral casing. The investigated turbine consists of a 1:11 scalemodel of the Kaplan turbine. The model runner has a diameter D = 500mm, the runner speed N = 595 rpm and thevolume flow Q = 0.522m 3/s. Operating conditions for investigation is close to the best efficiency operating point, at60% load.
On the present work, computation flow results must be indicated on the same location based on the experimentalinvestigation, specifically, the region focused to observe flow in the distributor, where tangential and axial velocities arerelevant.
Firstly, geometry has been redesigned on SolidWorks CAD software and inserted on CFD code as fluid domain,Fig. (1). Inlet and outlet boundary conditions are chosen following the three fluid turbulence models: (1) k ε, (2) k ω and (3) KE1E. (1) and (2) are two equations turbulence models and (3) is based on one equation model, Menter (1994).
Since in the open literature very few articles are available on computation analysis of this kind of geometry, thiswork aims to contribute to make CFD as a more popular tool among engineers because of its proposition of a goodnumerical solution that may sometimes replace older machine project methods.
Figure 1. Spiral casing of the Kaplan turbine
2. Governing Equations
Flow is considered to be viscous, incompressible and turbulent. Domain is discretized into small tetrahedralvolumes, as shown in Fig. (2) and special refinement around distributor surfaces, Fig. (3), were made for results
precision. Momentum and mass conservation equations have been used here as governing equations. These equationscan be written as the form shown bellow in Eq. (1) and Eq. (2) for mass and momentum conservation equations,respectively.
0=∇ur
(1)
Τ⋅∇+−∇= I p Dt
u D rr
ρ (2)
Where ρ is described as density, ur
is the velocity and its components are u, v and w for Cartesian directions x, y
and z, respectively, p is taken as pressure, I r
is the identity tensor, T as the stress tensor, defined in Eq. (3) and ρDu/Dtas the total derivative, as shown in Eq.(4). Equation (5) defines the strain tensor, S .
( ⋅∇−= utr uS T rr
3
1)(2µ ) (3)
( uu )t u
Dt u D rr
rr
⊗⋅∇+∂∂= ρ ρ ρ (4)
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[ uuuS T rrr∇+∇=
21
)( ] (5)
Figure 2. Spiral casing tetrahedral meshing. Figure 3. Meshing refinement around stay and guide vanes.
General governing equations are used for a basic knowledge of the code. Besides, some restrictions were considered for computational cost proposes, such as consider fluid flow incompressible and permanent.Turbulence models used by CFX 5.5.1 are showed in the section below, AEA Technology (2002).
2.1. Turbulence Models
2.1.1 The k – ε model
The k ε (k epsilon) model is the industry standard twoequation turbulence model. k is the turbulence kinetic energyand is defined as the variance of the fluctuations in velocity. It has dimensions of (L 2 T2), e.g. m 2/s2. e is the turbulence
eddy dissipation (the rate at which the velocity fluctuations dissipate) and has dimensions of k per unit time (L2
T3
), e.g.m2/s3.The k e model, like the zero equation model, is based on the eddy viscosity concept, so that in Eq. (6).
t ef µ µ µ += (6)
where µ ef is the effective viscosity accounting for turbulence, µ t is the turbulent viscosity. The k ε model assumes thatthe turbulence viscosity is linked to the turbulence kinetic energy and dissipation via the relation in Eq. (7).
ε ρ µ µ
2k ct
= (7)
where c µ is a constant with and its value is 0.09. The values of k and ε come directly from the differential transportequations for the turbulence kinetic energy and turbulence dissipation rate as shown in Eq. (8) and Eq. (9), respectively.
( ) ρε σ
µ ρ
ρ −=
∇⋅∇−⋅∇+
∂∂
k k
ef P k k U t k r
(8)
( ) ( ε ε ε ε
ρ ε
ε σ
µ ε ρ
ρε 21 c P c
k U
t k ef −=
∇⋅∇−⋅∇+
∂∂ r
) (9)
where the constants c ε1 = 1.44, c ε2 = 1.92, σk = 1 e σε= 1.3. P k is the shear production due to turbulence, which for incompressible flows is as Eq. (10).
( ) ( k U U U U U P t T
t k ρ µ µ +⋅∇⋅∇−∇+∇⋅∇=rrrrr
32 ) (10)
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2.1.2. The k – ω model
One of the advantages of the k ω formulation is the near wall treatment for lowReynolds number computations.The model does not involve the complex nonlinear damping functions required for the k ε model and is therefore moreaccurate and more robust. A lowReynolds number grid would require a near wall grid resolution of at least y+ < 2 .
This condition cannot be guaranteed in most applications and for this reason, a new near wall treatment wasdeveloped for the k ω model. It allows for smooth shift from a lowReynolds number form to a wall functionformulation.
The starting point of the present formulation is the k ω model developed by Wilcox (1986). It solves two transportequations, one for the turbulent kinetic energy, k , and one for the turbulent frequency, ω, respectively for Eq. (11) andEq. (12). The stress tensor is computed from the eddyviscosity concept.
( )
∇+⋅∇+−=⋅∇+
∂∂
k k P k U t k
k
t k σ
µ µ ω ρ β ρ
ρ '
r(11)
( ) ∇
+⋅∇+−=⋅∇+
∂∂
ω σ µ
µ βρω ω
α ω ρ ρω
ω
t k P
k U
t 2
r(12)
In addition to the independent variables, the density, ρ, and the velocity vector, U r
, are treated as known quantitiesfrom the NavierStokes method. In Equation (13), P k is the production rate of turbulence with the strain tensor S , as
shown in Eq. (5).
U S P t k
r∇= :2µ (13)
constants are given by 2=====ω σ σ β α β 2
403
95
0.09' k r
and the unknown Reynolds stress tensor, τ , is calculated from Eq. (14).
k S t δ ρ µ τ 32
2 −=r(14)
2.1.3. The one equation model – KE1E
This one equation model was developed by Menter (1994) which allows a transformation of the two equationturbulent viscosity models into one equation turbulence model. Menter followed Baldwin & Barth (1990) to describeone equation models and expressed the time derivative of the eddy viscosity by the times derivatives of k and ε, asshown in Eq. (15).
−=
Dt Dk
Dt Dk k
c Dt
v D t ε ε ε µ 2
2
2~
(15)
Equation (16) refers to the definition of the eddy viscosity:
ε µ
2~ k
cvt = (16)
and a single transport equation for the eddy viscosity, which, however, depends on k and ε as well as on the eddyviscosity was found in Eq. (17).
),,~(~
ε k v f Dt
v Dt
t = (17)
Since a complete set of equations, the one equation model can be derived by straightforward substitution, as shownin Eq. (18).
( )∇
+∇+−Ω=
t t
et t v
v E cv Dc
Dt v D ~
~~
~1211 σ
ν (18)
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Ω is the absolute vorticity and the constants of the model indicate the following values:
c1 = (c ε2  c ε1)cµ1/2 = 0.144 σ = σk = 1 c 2 = c 1κ 2 + σ1 = 1.86 κ = 0.41
ν ν ν ν
++=
t
t D ~1 A+ = 13.5
−
+−= κν
ν
At
e D
~
2 1
where c ε2, cε1 and c µ are the same values showed in the k ε model definition.
3. Flow survey and simulation
The following sections refer to the simulations done with the three turbulence models including experimentalresults. Since experimental investigation consists of LDV (Laser Doppler Velocimetry) measurements, a comparisonwith the simulations in the same section has been decided. Figure (4) shows the plane used for measurements as well asfor the simulation. Quantitative results were performed in a plot of velocities near the bend region in front of guidevanes. Figure (5) shows the velocity profile on the bend region. Results are available in mean tangential velocity andmean axial velocity.
LDV measurements are based in two components. Tangential velocity is normal to the measurement plane and theaxial velocity denotes the vertical velocity.
To observe the same velocities in the plane on the simulation, some equation expressions have been inserted intothe postprocessing software to calculate each component.
Figure 4. Top view of the direction of the measurement. Figure 5. Velocity profile at the bend region.
The experimental investigation used LDV measurements for validation of computations of the flow in thedistributor. The LDV technique uses the Doppler shift of reflected light particles and translates instantaneous velocity ina single point. Because of the characteristics of this kind of measurements, the laser beam needs transparent wall and
fluid to work correctly. Thus, a number of plexiglass windows have been installed in the water turbine model. Becauseof some difficulties to position the laser beam to get measurements, it was necessary to do them through the side of thespiral casing slightly nonorthogonal to the window (10.8°). By doing this, it was capable of making measurements asfar as possible into the distributor and avoids reflections from a guide vane. Additional description of the LDVtechnique is presented by Andersson (2000).
3.1. Results and discussion
The CFD code allows the insertion of additional information about restrictions to be made. Initial and boundaryconditions for velocities are inserted as following:
• Inlet: = mass flow = 522 kg/s at the intake;m&
• Outlet: p = relative pressure with respect with local ambient = 0 at the exit boundary.
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Figure 6. a Mean tangential velocity – k ε (m/s) b Mean axial velocity – k ε (m/s)
Figure 7. a Mean tangential velocity – k ω (m/s) b Mean axial velocity – k ω (m/s)
Figure 8. a Mean tangential velocity – KE1E (m/s) b Mean axial velocity – KE1E (m/s)
Figure 9. a Mean tangential velocity – experimental (m/s) b Mean axial velocity – experimental (m/s)
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0
50
100
150
200
250
300
0 2 3
Tangencial velocity(m/s)
y ( m m )
ke
measurements
0
50
100
150
200
250
300
1 0 1
Axial velocity(m/s)
y ( m m )
ke
measurements
Figure 10. k ε model and experimental results – velocity profile.
0
50
100
150
200
250
300
0 1 2 3
Tangencial Velocity (m/s)
y ( m m
)
kw
measurements
0
50
100
150
200
250
300
1 0 1
Axial Velocity (m/s)
y ( m m
)
kw
measurements
Figure 11. k ω and experimental results – velocity profile.
0
50
100
150
200
250
300
0 1 2 3
Tangencial velocity (m/s)
y ( m m
)
KE1E
measurements
0
50
100
150
200
250
300
1 0 1
Axial velocity(m/s)
y ( m m
)
KE1E
measurements
Figure 12. KE1E  one equation model and experimental results – velocity profile.
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In the spiral casing, treedimensional tetrahedral meshing has been formed. Meshing generation can be described asthe whole fluid domain and the distributor components as no slip wall condition. In order to determine a steady statevalue, a 10 5 convergence criterion has been chosen with predefined timescale control for each turbulence model(484096 nodes and 2621366 elements). For k ε and k ω models, CPU time required to solve the numerical problem wasabout 3 hours and for the one equation model (KE1E), about 2 hours on a 2GB RAM, 2.0 GHz Pentium 4 Dell 340Precision. See table 1 for CPU time detailed information.
Table 1. Turbulence models outer loop iteration numbers and convergence CPU time.
Turbulence Model Number of Iterations CPU time(s)
kε 41 11242.87kω 41 11220.75
KE1E 18 7049.51
Simulation shows contour of the tangential velocity in the local measurement plane, for the three turbulence modelsin Fig. (6.a), Fig. (7.a) and Fig. (8.a) had almost same gradient velocity behavior. A velocity increase at the lower ringdue to the smaller area at the distributor region has been detected. Also a stagnation region is located at the front of theguide vanes. In experimental measurements, according to H. Nilsson et all (2001), the main features of the flow in thedistributor are that the tangential velocity component increases at the lower ring, as shown in Fig. (9.a), which is aneffect of the sharp bend of the flow after the distributor. A stagnation region occurs in front of a guide vane, a guidevane wake region is present at the radially innermost of the measurements and there is a sharp increase in the magnitudeof the axial velocity component close to the bend of the flow, as seen on Fig. (9.b).
In both type of results, simulation and experimental measurements, similar velocity contours can be seen. At theouter region of the spiral casing, flow seems to be slightly slower than experimental results. Simulation detectstangential velocity increase also from the top of the spiral casing to the outlet region.
Figure (6.b), Fig. (7.b) and Fig. (8.b) show mean axial velocity contour at the same measurement plane. A velocityincrease is detected at the bend region. It decreases at the guide vane entrance and at the inner section of the plane,velocity suddenly increases due to the runner entrance. Simulations agree with experimental results concerning to thevelocity increase at the bend region and flow behavior.
Figure (10), Fig. (11) and Fig. (12) plot the tangential and axial velocities distributions in a profile at the bendregion, as seen on the highlighted red line in Fig. (5). Results show good agreement with those presented by H. Nilssonet all (2001) measurements.
Figure (15) shows streamline velocity from the intake to the spiral casing exit. Flow structure is almost radial at theexit region (after the stay and guide vanes region). Figure (13) shows velocity vectors at a horizontal midplane. Figure(14) shows vectors closely at the guide vane passage. Fluid contours the vane surface avoiding recirculation from thedistributor entrance and rotating flow for runner at the outlet region is distributed circumferentially. Finally, Fig. (16)shows axial at the spiral casing outlet. Velocity is almost well distributed considering variation around the componentwall.
Concerning computing cost and time considerations, the one equation model solved successfully the simulationshowing similar results and it was calculated in a few numbers of iterations for convergence comparing the two other models.
Figure 13. Velocity vectors projection at the horizontal midplane. Figure 14. A closer look at the velocity vectors
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Figure 15. Plot of streamline velocity
Figure 16. Axial velocity at the spiral casing outlet.
4. Conclusions
The flow structure simulation using three different turbulence models to compare with experimental results have been obtained using a finite volume based CFD tool with tetrahedral meshing generated for analysis. Results show goodqualitative and quantitative agreement with measurements and all the models successfully analyzed the threedimensional flow at an extremely complex geometry hydro machine component at the same experimental operatingcondition. The CFD code contributes to develop design efficiently hydraulic turbines which are a field that lackshydrodynamics theory application.
5. References
AEA Technology, 2002, “Introduction to CFX 5 version 5.5.1”, United Kingdom, 276 p.Andersson, U. and Karlsson, R., 2000, “Quality aspects of the Turbine 99 draft tube experiment”. Proceedings from
Turbine 99 – workshop on draft tube flow, ISSN: 14021536.Gros, L., Kueny, J.L., Avellan, F. and Bellet, L., 1998, “Numerical flow analysis of the GAMM turbine at nominal and
offdesign operating conditions”, Proceedings of the XIX IAHR Symposium, Hydraulic Machinery and Cavitation, pp.121128.
Kurokawa, J. and Nagahara, H., 1986, “Flow characteristics in spiral casings in water turbines”, IAHR Symp.,Montreal, Canada, Vol. 2, Paper No. 62.
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Maji, P.K. and Biswas, G., 1998, “Threedimensional analysis of flow in the spiral casing of a reaction turbine using adifferently weighted Petrov Galerkin method”, Computer Methods in Applied Mechanics and Engineering, Vol.167, pp. 167190.
Maji, P.K. and Biswas, G., 1999, “Analysis of flow in the spiral casing using a streamline upwind Petrov Galerkinmethod”, International Journal for Numerical Methods in Engineering, Vol. 45, pp. 147174.
Menter, F.R., 1994, “Eddy Viscosity Transport Equations and their Relation to the k ε Model”, ASME Journal of FluidsEngineering, Vol. 119, pp. 876884.
Nilsson, H., Andersson, U. and Videhult, S., 2001, “An experimental investigation of the flow in the spiral casing anddistributor of the Hölleforsen Kaplan turbine model”, Publication 01/05, Chalmers University of Technology,Göteborg, Sweden, 12 p.
Nilsson, H. and Davidson, L., 2001, “A validation of parallel multiblock CFD against the GAMM Francis water turbinerunner at best efficiency and offdesign operating conditions”, Publication 02/05, Chalmers University of Technology, Göteborg, Sweden, 42 p.